Establish planar localization at mirror density p=2/5

Establish whether the Manhattan mirror model at mirror density p=2/5 exhibits planar localization, equivalently whether every trajectory is almost surely periodic or otherwise confined in the infinite plane.

Background

The paper develops a boundary-parity observable and proves a sufficient finite-scale criterion implying almost-sure periodicity of all trajectories. At p=2/5, however, the certified finite-size probabilities remain far below the threshold 0.8457 required by that criterion.

Consequently, the computations establish finite-volume nonmonotonicity but do not settle the broader localization question for the planar Manhattan mirror model at this density.